On classes of estimators of the variance function of a linear estimator

Summary The variance function of a linear estimator can be expressed into a quadratic form. The present paper presents classes of estimators of this quadratic form along the lines implicitly suggested byHorvitz andThompson [1952] while formulating the classes of linear estimators. Accordingly it is noted that there exist nine principal classes of estimators out of which one principal class is examined in detail. Furthermore to illustrate the theory an example is considered where the expression for a unique estimator variance of the best estimator in theT ₁ class is derived.     

Multivariate Hill Estimators

We propose two classes of semi‐parametric estimators for the tail index of a regular varying elliptical random vector. The first one is based on the distance between a tail probability contour and the observations outside this contour. We denote it as the class of separating estimators. The second one is based on the norm of an arbitrary order. We denote it as the class of angular estimators. We show the asymptotic properties and the finite sample performances of both classes. We also illustrate the separating estimators with an empirical application to 21 worldwide financial market indexes.     

A remark on Lehmann-unbiased estimators for scale resp. Location parameters

Summary In this paper Lehmann-unbiased estimation of the scale and location parameter is considered. Lehmann-unbiased estimators depend strongly on the form of the loss function. Therefore quadratic and the other loss functions are discussed. Results of this paper, obtained in the class of linear statistics, can be specified to these obtained byGoodman andKiciska-Slaby [1982a, 1982b].     

Estimation of the characteristic roots of the scale matrix

The simultaneous estimation of the characteristic roots of the scale matrix of the multivariatet-model is considered. The improved estimation strategies are developed in the light of a quadratic loss function. It is demonstrated analytically and numerically that the class of proposed estimators outperforms the class of usual estimators in the sense of having smaller risk.     

Quadratic Engel Curves with Measurement Error: Evidence from a Budget Survey*

This paper examines the importance of accounting for measurement error in total expenditure in the estimation of Engel curves, based on the 1994 Ethiopian Urban Household Survey. Using Lewbel's [Review of Economics and Statistics (1996 ), Vol. 78, pp. 718–725] estimator for demand models with correlated measurement errors in the dependent and independent variables, we find robust evidence of a quadratic relationship between food share and total expenditure in the capital city, and significant biases in various estimators that do not correct for correlated measurement errors.     

Iterative Density Estimation from Contaminated Observations

We introduce an iterative procedure for estimating the unknown density of a random variable X from n independent copies of Y=X+ɛ, where ɛ is normally distributed measurement error independent of X. Mean integrated squared error convergence rates are studied over function classes arising from Fourier conditions. Minimax rates are derived for these classes. It is found that the sequence of estimators defined by the iterative procedure attains the optimal rates. In addition, it is shown that the sequence of estimators converges exponentially fast to an estimator within the class of deconvoluting kernel density estimators. The iterative scheme shows how, in practice, density estimation from indirect observations may be performed by simply correcting an appropriate ordinary density estimator. This allows to assess the effect that the perturbation due to contamination by ɛ has on the density to be estimated. We also suggest a method to select the smoothing parameter required by the iterative approach and, utilizing this method, perform a simulation study.     

Jack-knifing the ratio and the product estimators in double sampling

Summary Almost unbiased ratio and product type estimators have been obtained with the help of the Jack-Knifing technique for simple random sampling in two phases. The mean square errors of the resulting estimators have been compared with those of the corresponding usual (biased) estimators and it has been found that they are approximately same. This study generalizes similar single sampling results ofDurbin [1959],Shukla [1976] and others.     

Estimation of the precision matrix of multivariate Pearson type II model

In this paper, the problem of estimating the precision matrix of a multivariate Pearson type II-model is considered. A new class of estimators is proposed. Moreover, the risk functions of the usual and the proposed estimators are explicitly derived. It is shown that the proposed estimator dominates the MLE and the unbiased estimator, under the quadratic loss function. A simulation study is carried out and confirms these results. Improved estimator of tr (Σ ⁻¹) is also obtained.     

Almost unbiased product-type estimator

In sample survey methods the use of product estimators was suggested byMurthy [1964] andSrivastava [1966] and were found to serve good purpose provided the two variables viz. the main variable under study and the auxiliary variable have a very high negative correlation between them. The product estimators suggested by them are biased. In the present paper the author has obtained unbiased product estimators (to the first degree of approximation) with the help of the technique developed byQuenouille [1956] and has established that this new estimator is better than the other product estimator in the mean square error sense.     

Penalized Quantile Regression with Semiparametric Correlated Effects: An Application with Heterogeneous Preferences

This paper proposes new ?₁‐penalized quantile regression estimators for panel data, which explicitly allows for individual heterogeneity associated with covariates. Existing fixed‐effects estimators can potentially suffer from three limitations which are overcome by the proposed approach: (i) incidental parameters bias in nonlinear models with large N and small T ; (ii) lack of efficiency; and (iii) inability to estimate the effects of time‐invariant regressors. We conduct Monte Carlo simulations to assess the small‐sample performance of the new estimators and provide comparisons of new and existing penalized estimators in terms of quadratic loss. We apply the technique to an empirical example of the estimation of consumer preferences for nutrients from a demand model using a large transaction‐level dataset of household food purchases. We show that preferences for nutrients vary across the conditional distribution of expenditure and across genders, and emphasize the importance of fully capturing consumer heterogeneity in demand modeling. Copyright © 2016 John Wiley & Sons, Ltd.     

An efficient estimator using auxiliary information

The use of ratio and product methods of estimation using auxiliary information for estimating the mean of a finite population is well known.Srivastava [1967] andReddy [1973] proposed ratio-cum-product type estimators. This paper proposes a transformed estimator which is even more efficient than these estimators for a wide range of the value of the correlation coefficient between the main and auxiliary variables.     

Unbiased product estimators

Summary The paper presents a comparative study of product estimators proposed byRobson [1957] andMurthy [1964]. It is seen that the Robson's estimator gives a better performance.     

On unordering Horvitz and Thompson'sT 1 class of linear estimators

Summary When elements of a finite population are sampled with varying probability selection at each draw,Horvitz andThompson [1952] have formulated certain classes of linear estimators to bear on the problem of providing a smaple appraisal of the population total.Horvitz andThompson's T ₁ class is an ordered one, which was examined by the present author [1967 b]. For some sampling procedures a best estimator exists for theT ₁ class. Subsequently the present author [1967 c] appliedMurthy's technique [Murthy 1967] of unordering an ordered estimator and derived a more efficient estimator. The present paper is concerned with applyingMurthy's technique to theT ₁ class itself, and examining the unorderedT ₁ class. Curiously enough, it is noted that the condition of unbiasedness is sufficient to completely specify the unorderedT ₁ class for the sampling procedure considered here.Research sponsored by Marathwada University, Aurangabad, India; under Grant No. Research-12-68-69/3314-16.     

On the existence of moments of partially restricted reduced form coefficients

In this paper ridgelike Bayesian estimators of structural coefficients have been used to form the partially restricted reduced form estimators. These partially restricted reduced form estimators are simple in form and possess finite sampling moments and risk in contrast to other restricted reduced form estimators that possess no finite moments and have infinite risk relative to quadratic loss functions. The usual k-class implied partially restricted reduced form estimators with 0≦k≦1 do not posses finite moments unless the degree of overidentification (or the excess of sample size over the number of coefficients) of the structural equation being estimated is suitably restricted.     

Data-based ranking of realised volatility estimators

This paper presents new methods for comparing the accuracy of estimators of the quadratic variation of a price process. I provide conditions under which the relative accuracy of competing estimators can be consistently estimated (as T→∞), and show that forecast evaluation tests may be adapted to the problem of ranking these estimators. The proposed methods avoid making specific assumptions about microstructure noise, and facilitate comparisons of estimators that would be difficult using methods from the extant literature, such as those based on different sampling schemes. An application to high frequency IBM data between 1996 and 2007 illustrates the new methods.     

Best unbiased estimators for the parameters of a two-parameter Pareto distribution

Summary For a two-parameter Pareto distributionMalik [1970] has shown that the maximum likelihood estimators of the parameters are jointly sufficient. In this article the maximum likelihood estimators are shown to be jointly complete. Furthermore, unbiased estimators for the two parameters are obtained and are shown to be functions of the jointly complete sufficient statistics, thereby establishing them as the best unblased estimators of the two parameters.This research is a part of the first author's Ph.D. dissertation. The authors wish to thank Dr. Kenny S. Crump, for many helpful suggestions and a referee for improvements in the proofs.     

A Note on the Theme of Too Many Instruments*

The difference and system generalized method of moments (GMM) estimators are growing in popularity. As implemented in popular software, the estimators easily generate instruments that are numerous and, in system GMM, potentially suspect. A large instrument collection overfits endogenous variables even as it weakens the Hansen test of the instruments’ joint validity. This paper reviews the evidence on the effects of instrument proliferation, and describes and simulates simple ways to control it. It illustrates the dangers by replicating Forbes [American Economic Review (2000) Vol. 90, pp. 869–887] on income inequality and Levine et al. [Journal of Monetary Economics] (2000) Vol. 46, pp. 31–77] on financial sector development. Results in both papers appear driven by previously undetected endogeneity.     

Variance estimation for unequal probability sampling

In this paper, we discuss the optimality of the variance estimator of the Horvitz-Thompson estimator proposed by Kott (1988) in the class of model-unbiased quadratic estimators. We also propose some improved estimators over Kott's estimator in the class of general quadratic estimators. Received: February 1999     

On estimating nonnegative definite quadratic forms

It is often required to estimate a quadratic form in survey sampling, especially when one has to estimate the mean squared error of a linear estimator of the population total. In this note we consider the problem of obtaining uniformly nonnegative quadratic unbiased estimators for nonnegative definite quadratic forms. The estimators considered here are necessarily quadratic. Received January 1997     

Pre-testing for linear restrictions in a regression model with spherically symmetric disturbances

In this paper we derive the exact risk (under quadratic loss) of pre-test estimators of the prediction vector and of the error variance of a linear regression model with spherically symmetric disturbances. The pre-test in question is one of the validity of a set of exact linear restrictions on the model's coefficient vector. We demonstrate how the known results for the model with normal disturbances can be extended to this broader case. We also show that the critical value of unity results in a minimum of the risk of the pre-test estimator of the error variance. To illustrate the results we assume multivariate Student-t regression disturbances and numerically evaluate the derived expressions.